We introduce Prob-GParareal, a probabilistic extension of the GParareal algorithm designed to provide uncertainty quantification for the Parallel-in-Time (PinT) solution of (ordinary and partial) differential equations (ODEs, PDEs). The method employs Gaussian processes (GPs) to model the Parareal correction function, in line with GParareal, further enabling the propagation of numerical uncertainty across time and yielding probabilistic forecasts of the system's evolution. Furthermore, Prob-GParareal accommodates probabilistic initial conditions and maintains compatibility with classical numerical solvers, ensuring its straightforward integration into existing Parareal frameworks. Here, we first conduct a theoretical analysis of the computational complexity and derive error bounds of Prob-GParareal. Then, we numerically demonstrate the accuracy and robustness of the proposed algorithm on five benchmark ODE systems, including chaotic, stiff, and bifurcation problems. To showcase the flexibility and potential scalability of the proposed algorithm, we also consider Prob-nnGParareal, a variant obtained by replacing the GPs in Parareal with the nearest-neighbors GPs, illustrating its increased performance on an additional PDE example. This work bridges a critical gap in the development of probabilistic counterparts to established PinT methods.

翻译：我们引入了Prob-GParareal，这是GParareal算法的概率扩展，旨在为（常微分方程和偏微分方程）的并行时间（PinT）求解提供不确定性量化。该方法采用高斯过程（GPs）对Parareal校正函数进行建模，与GParareal一致，进一步实现了数值不确定性在时间上的传播，并生成系统演化的概率预测。此外，Prob-GParareal支持概率初始条件，并保持与经典数值求解器的兼容性，确保其能无缝集成到现有的Parareal框架中。在此，我们首先对Prob-GParareal的计算复杂度进行理论分析，并推导其误差界。然后，我们在五个基准常微分方程系统（包括混沌、刚性和分岔问题）上数值验证了所提算法的准确性和鲁棒性。为了展示所提算法的灵活性和潜在可扩展性，我们还考虑了Prob-nnGParareal，这是通过将Parareal中的高斯过程替换为最近邻高斯过程得到的变体，并在一个额外的偏微分方程实例上展示了其性能提升。这项工作填补了为现有并行时间方法开发概率对应物方面的关键空白。